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December 8th, 2017, 07:10 PM   #1
Joined: Apr 2011

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Collatz problem


I got excited when I understood the Collatz conjecture.
I have some ideas that I want to try out but I am stuck on where to begin. I allready know some fundamentals, but want to learn more math to analyse the numbers related to the Collatz function. But, that in regards to base-2 numbers. I have background in programming, binary numbers and a little on Cellular Automata allready. I took elementary college course on derivation and integration, but have found no use of it yet.

Im looking for ideas, talks and discussions and the like about the Collatz conjecture and its issues. Also willing to learn more math on this issue.

I saw allready someone had started a thread about this, but ill just repeat how the Collatz function is defined:

C(x) = if x=odd then { 3x+1 } else
if x=even then { x/2 }

And so the collatz conjecture states that any initial number x will eventually reach 1 (or cycle: 1,4,2,1,4,2,...).

I found out that if C(x) reaches a pow2 number (that is any 2^n number: 1,2,4,8,16,32,64,..etc) then it will automatically reach 1. And after googling it, I saw that that was allready known information of course.

So looking at the bits of how these C(x) numbers behave, its almost like a Galois pseudorandom number generator (which is mostly about polynomials).

And something that people have figured out (correct me if I am wrong) is that
the odd part of the condition is changed to {(3x+1)/2} also shortcuts the branches to reach the number 1, but if it is proven someway it will also prove the original conjecture.

Anyways, is it possible to prove something by entropy of binary numbers like taking the hamming weight (or some other trick) of the bitstring that the C(x) function creates?

And the patterns that often occurs in one of the numbers is the ...100000001 pattern and the stuff inbetween just permutates into something like (pow2)....10000... and shifts to the right.

Wikipedia have a nice paragraph about how to do this with bit-manipulation.
Anyways, ill stop rambling and ask if people have some thoughts and ideas on the patterns made by the Collatz function and analysis-methods (preferably in base-2) like discreet methods.

Thanks for your time.

(Also i posted a similar idea in mathexchange forum but just got flamed all the way.. Im not out after wannabe-mathematicians to downvote a thread and say that I have not solved the Collatz-problem, because I have never claimed that I have. Im just out after collaboration and im willing to learn more math, I want creative responses rather than: what do you mean by this and that)..
Rudi is offline  
April 22nd, 2018, 06:28 PM   #2
Joined: Sep 2013
From: Reno,NV

Posts: 14
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Math Focus: Number Theory
since $2^{2k} - 1$ is odd and always divisible by 3, a generation of these numbers always results in 1. when encountering 1,5,21,85,341, we define second problem why do we generate those numbers. Just my 2 cents worth...
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